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Groups, Geometry, and Dynamics

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Volume 14, Issue 3, 2020, pp. 1023–1042
DOI: 10.4171/GGD/573

Published online: 2020-10-21

On localizations of quasi-simple groups with given countable center

Ramón Flores[1] and José L. Rodríguez[2]

(1) Universidad de Sevilla, Spain
(2) Universidad de Almería, Spain

A group homomorphism $i\colon H \to G$ is a localization of $H$, if for every homomorphism $\varphi\colon H\to G$ there exists a unique endomorphism $\psi\colon G\to G$ such that $i \psi=\varphi$ (maps are acting on the right). Göbel and Trlifaj asked in [18, Problem 30.4(4), p. 831] which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable abelian group is indeed the center of some localization of a quasi-simple group, i.e., a central extension of a simple group. The proof uses Obraztsov and Ol'shanskii's construction of infinite simple groups with a special subgroup lattice and also extensions of results on localizations of finite simple groups by the second author and Scherer, Thévenaz and Viruel.

Keywords: Localization, simple group, quasi-simple group, abelian group

Flores Ramón, Rodríguez José: On localizations of quasi-simple groups with given countable center. Groups Geom. Dyn. 14 (2020), 1023-1042. doi: 10.4171/GGD/573